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Physics 111: Mechanics Lecture 2 Wenda Cao NJIT Physics Department

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September 8, 2008 Motion along a straight line Motion Position and displacement Average velocity and average speed Instantaneous velocity and speed Acceleration Constant acceleration: A special case Free fall acceleration

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September 8, 2008 Motion Everything moves! Motion is one of the main topics in Physics 111 Simplification: Moving object is a particle or moves like a particle: “point object” Simplest case: Motion along straight line, 1 dimension LAX Newark

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September 8, 2008 One Dimensional Position x What is motion? Change of position over time. How can we represent position along a straight line? Position definition: Defines a starting point: origin (x = 0), x relative to origin Direction: positive (right or up), negative (left or down) It depends on time: t = 0 (start clock), x(t=0) does not have to be zero. Position has units of [Length]: meters. x = m x = - 3 m

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September 8, 2008 Vector and Scalar A vector quantity is characterized by having both a magnitude and a direction. Displacement, Velocity, Acceleration, Force … Denoted in boldface type with an arrow over the top. A scalar quantity has magnitude, but no direction. Distance, Mass, Temperature, Time … For the motion along a straight line, the direction is represented simply by + and – signs. + sign: Right or Up. - sign: Left or Down. 2-D and 3-D motions.

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September 8, 2008 Quantities in Motion Any motion involves three concepts Displacement Velocity Acceleration These concepts can be used to study objects in motion.

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September 8, 2008 Displacement Displacement is a change of position in time. Displacement: f stands for final and i stands for initial. It is a vector quantity. It has both magnitude and direction: + or - sign It has units of [length]: meters. x 1 (t 1 ) = m x 2 (t 2 ) = m Δx = -2.0 m m = -4.5 m x 1 (t 1 ) = m x 2 (t 2 ) = m Δx = +1.0 m m = +4.0 m

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September 8, 2008 Distance and Position-time graph Displacement in space From A to B: Δx = x B – x A = 52 m – 30 m = 22 m From A to C: Δx = x c – x A = 38 m – 30 m = 8 m Distance is the length of a path followed by a particle from A to B: d = |x B – x A | = |52 m – 30 m| = 22 m from A to C: d = |x B – x A |+ |x C – x B | = 22 m + |38 m – 52 m| = 36 m Displacement is not Distance.

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September 8, 2008 Velocity Velocity is the rate of change of position. Velocity is a vector quantity. Velocity has both magnitude and direction. Velocity has a unit of [length/time]: meter/second. Definition: Average velocity Average speed Instantaneous velocity

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September 8, 2008 Average Velocity Average velocity It is slope of line segment. Dimension: [length/time]. SI unit: m/s. It is a vector. Displacement sets its sign.

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September 8, 2008 Average Speed Average speed Dimension: [length/time], m/s. Scalar: No direction involved. Not necessarily close to V avg : S avg = (6m + 6m)/(3s+3s) = 2 m/s V avg = (0 m)/(3s+3s) = 0 m/s

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September 8, 2008 Graphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions. It is a vector quantity. An object moving with a constant velocity will have a graph that is a straight line.

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September 8, 2008 Instantaneous Velocity Instantaneous means “at some given instant”. The instantaneous velocity indicates what is happening at every point of time. Limiting process: Chords approach the tangent as Δt => 0 Slope measure rate of change of position Instantaneous velocity: It is a vector quantity. Dimension: [Length/time], m/s. It is the slope of the tangent line to x(t). Instantaneous velocity v(t) is a function of time.

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September 8, 2008 Uniform velocity is constant velocity The instantaneous velocities are always the same, all the instantaneous velocities will also equal the average velocity Begin with then Uniform Velocity x x(t) t0 xixi xfxf v v(t) t0 tftf vxvx titi

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September 8, 2008 Average Acceleration Changing velocity (non-uniform) means an acceleration is present. Acceleration is the rate of change of velocity. Acceleration is a vector quantity. Acceleration has both magnitude and direction. Acceleration has a unit of [length/time 2 ]: m/s 2. Definition: Average acceleration Instantaneous acceleration

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September 8, 2008 Average Acceleration Average acceleration Velocity as a function of time When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph

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September 8, 2008 Instantaneous and Uniform Acceleration The limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be uniform. The instantaneous acceleration will be equal to the average acceleration Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

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September 8, 2008 Relationship between Acceleration and Velocity Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

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September 8, 2008 Relationship between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

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September 8, 2008 Relationship between Acceleration and Velocity Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

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September 8, 2008 Kinematic Variables: x, v, a Position is a function of time: Velocity is the rate of change of position. Acceleration is the rate of change of velocity. Position Velocity Acceleration Graphical relationship between x, v, and a An elevator is initially stationary, then moves upward, and then stops. Plot v and a as a function of time.

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September 8, 2008 Motion with a Uniform Acceleration Acceleration is a constant Kinematic Equations

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September 8, 2008 Notes on the Equations Given initial conditions: a(t) = constant = a, v(t=0) = v 0, x(t=0) = x 0 Start with We have Shows velocity as a function of acceleration and time Use when you don’t know and aren’t asked to find the displacement

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September 8, 2008 Given initial conditions: a(t) = constant = a, v(t=0) = v 0, x(t=0) = x 0 Start with Since velocity change at a constant rate, we have Gives displacement as a function of velocity and time Use when you don’t know and aren’t asked for the acceleration Notes on the Equations

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September 8, 2008 Given initial conditions: a(t) = constant = a, v(t=0) = v 0, x(t=0) = x 0 Start with We have Gives displacement as a function of time, initial velocity and acceleration Use when you don’t know and aren’t asked to find the final velocity Notes on the Equations

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September 8, 2008 Given initial conditions: a(t) = constant = a, v(t=0) = v 0, x(t=0) = x 0 Start with We have Gives velocity as a function of acceleration and displacement Use when you don’t know and aren’t asked for the time Notes on the Equations

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September 8, 2008 Problem-Solving Hints Read the problem Draw a diagram Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations Label all quantities, be sure all the units are consistent Convert if necessary Choose the appropriate kinematic equation Solve for the unknowns You may have to solve two equations for two unknowns Check your results Estimate and compare Check units

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September 8, 2008 Free Fall Acceleration Earth gravity provides a constant acceleration. Most important case of constant acceleration. Free-fall acceleration is independent of mass. Magnitude: |a| = g = 9.8 m/s 2 Direction: always downward, so a g is negative if define “up” as positive, a = -g = -9.8 m/s 2 Try to pick origin so that x i = 0 y

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September 8, 2008 Free Fall Acceleration Two important equation: Begin with t 0 = 0, v 0 = 0, x 0 = 0 So, t 2 = 2|x|/g same for two balls! Assuming the leaning tower of Pisa is 150 ft high, neglecting air resistance, t = (2 150 0.305/9.8) 1/2 = 3.05 s x 0

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September 8, 2008 Summary This is the simplest type of motion It lays the groundwork for more complex motion Kinematic variables in one dimension Positionx(t)mL Velocityv(t)m/sL/T Accelerationa(t)m/s 2 L/T 2 All depend on time All are vectors: magnitude and direction vector: Equations for motion with constant acceleration: missing quantities x – x 0 v t a v 0

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